Smooth Toric Deligne-Mumford Stacks Barbara Fantechi, Etienne Mann, Fabio Nironi

Smooth toric Deligne-Mumford stacks Barbara Fantechi, Etienne Mann, Fabio Nironi

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Barbara Fantechi, Etienne Mann, Fabio Nironi. Smooth toric Deligne-Mumford stacks. Jour- nal für die reine und angewandte Mathematik, Walter de Gruyter, 2010, 648, pp.201-244. 10.1515/CRELLE.2010.084. hal-00166751v2

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J. reine angew. Math. 648 (2010), 201—244 Journal fu¨r die reine und DOI 10.1515/CRELLE.2010.084 angewandte Mathematik ( Walter de Gruyter Berlin New York 2010

Smooth toric Deligne-Mumford stacks

By Barbara Fantechi at Trieste, Etienne Mann at Montpellier and Fabio Nironi at New York

Abstract. We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a ‘‘torus. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

Contents

Introduction 1. Notations and background 1.1. Conventions and notations 1.2. Smooth Deligne-Mumford stacks and orbifolds 1.3. Root constructions 1.4. Rigidification 1.5. Diagonalizable group schemes 1.6. Toric varieties 1.7. Picard stacks and action of a Picard stack 2. Deligne-Mumford tori 3. Definition of toric Deligne-Mumford stacks 4. Canonical toric Deligne-Mumford stacks 4.1. Canonical smooth Deligne-Mumford stacks 4.2. The canonical stack of a simplicial toric variety 5. Toric orbifolds 6. Toric Deligne-Mumford stacks 6.1. Gerbes and root constructions 6.2. Gerbes on toric orbifolds 6.3. Characterization of a toric Deligne-Mumford stack as a gerbe over its rigidification 7. Toric Deligne-Mumford stacks versus stacky fans 7.1. Toric Deligne-Mumford stacks as global quotients 7.2. Toric Deligne-Mumford stacks and stacky fans 7.3. Examples

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202 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Appendix A. Uniqueness of morphisms to separated stacks Appendix B. Action of a Picard stack Appendix C. Stacky version of Zariskis Main Theorem References

Introduction

A toric variety is a normal, separated variety X with an open embedding T , X of a torus such that the action of the torus on itself extends to an action on X. To a toric! variety one can associate a fan, a collection of cones in the lattice of one-parameter subgroups of T. Toric varieties are very important in algebraic geometry, since algebro-geometric properties of a toric variety translate in combinatorial properties of the fan, allowing to test con- jectures and produce interesting examples.

In [10] Borisov, Chen and Smith define toric Deligne-Mumford stacks as explicit global quotient (smooth) stacks, associated to combinatorial data called stacky fans. Later, Iwanari proposed in [22] a definition of toric triple as an orbifold with a torus action having a dense orbit isomorphic to the torus1) and he proved that the 2-category of toric triples is equivalent to the 2-category of ‘‘toric stacks (We refer to [22] for the definition of ‘‘toric stacks). Nevertheless, it is clear that not all toric Deligne-Mumford stacks are toric triples, since some of them are not orbifolds.

Then the generalization of the D-collections deﬁned for toric varieties by Cox in [14] was done by Iwanari in [23] in the orbifold case and by Perroni in [31] in the general case.

In this paper, we define a Deligne-Mumford torus T as a Picard stack isomorphic to T BG, where T is a torus, and G is a finite abelian group; we then define a smooth toric Deligne-Mumford stack as a smooth separated Deligne-Mumford stack with the action of a Deligne-Mumford torus T having an open dense orbit isomorphic to T. We prove a classification theorem for smooth toric Deligne-Mumford stacks and show that they coin- cide with those defined by [10].

The ﬁrst main result of this paper is a bottom-up description of smooth toric Deligne- Mumford stacks, as follows: the structure morphism X X to the coarse moduli space factors canonically via the toric morphisms !

X Xrig Xcan X ! ! ! where X Xrig is an abelian gerbe over Xrig; Xrig Xcan is a ﬁbered product of roots of toric divisors;! and Xcan X is the minimal orbifold! having X as coarse moduli space. Here X is a simplicial toric! variety, and Xrig and Xcan are smooth toric Deligne-Mumford stacks. More precisely, this bottom up construction can be stated as follows.

Theorem I. Let X be a smooth toric Deligne-Mumford stack with Deligne-Mumford torus isomorphic to T BG. Denote by X the coarse moduli space of X. Denote by n the number of rays of the fan of X.

1) For the meaning of orbifold in this paper, see §1.2.

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 203

n rig (1) There exist unique a1; ...; an A N>0 such that the stack X is isomorphic, as toric Deligne-Mumford stack, to

a1 an can Xcan can can can Xcan D1 = X X Dn = ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi can where Di is the divisor corresponding to the ray ri.

l rig ... l X X (2) Given G mbj . There exist L1; ; L in Pic such that is isomorphic, as j 1 Q toric Deligne-Mumford stack, to

b1 rig bl rig L1=X X rig X rig Ll=X : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rig rig Moreover, for any j A 1; ...; l , the class Lj in Pic X =bj Pic X is unique. f g In the process, we get a description of the Picard group of smooth toric Deligne- Mumford stacks, which allows us to characterize weighted projective stacks as complete toric orbifolds with cyclic Picard group (cf. Proposition 7.28). Moreover, we classify all complete toric orbifolds of dimension 1 (cf. Example 7.31). We also show that the natural map from the Brauer group of a smooth toric Deligne-Mumford stack with trivial generic stabilizer to its open dense torus is injective (cf. Theorem 6.11).

The second main result of this article is to give an explicit relation between the smooth toric Deligne-Mumford stacks and the stacky fans.

Theorem II. Let X be a smooth toric Deligne-Mumford stack with coarse moduli space the toric variety denoted by X. Let S be a fan of X in NQ : N nZ Q. Assume that the rays of S generate NQ. There exists a stacky fan such that X is isomorphic, as toric Deligne-Mumford stack, to the smooth Deligne-Mumford stack associated to the stacky fan. Moreover, if X has a trivial generic stabilizer then the stacky fan is unique.

When the smooth toric Deligne-Mumford stack X has a generic stabilizer the non- uniqueness of the stacky fan comes from three di¤erent choices. We refer to Remark 7.26 for a more precise statement. This result gives a geometrical interpretation of the combinatorial data of the stacky fan. In fact, the stacky fan can be read o¤ the geometry of the smooth toric Deligne-Mumford stack just like the fan can be read o¤ the geometry of the toric variety. Notice that one can deduce the above theorem when X is an orbifold from [31], Theorem 2.5, and [23], Theorem 1.4, and the geometric characterization of [24], Theorem 1.3.

In the first part of this article, we fix the conventions and collect some results on smooth Deligne-Mumford stacks, root constructions, rigidification, toric varieties, Picard stacks and the action of a Picard stack. In Section 2, we define Deligne-Mumford tori. Sec- tion 3 contains the definition of smooth toric Deligne-Mumford stacks. In Section 4, we first define canonical smooth Deligne-Mumford stacks and then we show that the canonical stack associated to a simplicial toric variety is a smooth toric Deligne-Mumford stack (cf. Theorem 4.11). In Section 5, we prove the first part of Theorem I. In Section 6, we first prove in Proposition 6.9 that the essentially trivial banded gerbes over X are in bijection

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204 Fantechi, Mann and Nironi, Deligne-Mumford stacks with ﬁnite extensions of the Picard group of X; then, we show that the natural map from the Brauer group of a smooth toric Deligne-Mumford stack with trivial generic stabilizer to its open dense torus is injective (cf. Theorem 6.11). Finally, we prove the second statement of Theorem I. In Section 7, we prove Theorem II and give some explicit examples. In Appendix B, we have put some details about the action of a Picard stack.

Acknowledgments. The authors would like to acknowledge support from IHP, Mittag-Le¿er Institut, SNS where part of this work was carried out, as well as the Euro- pean projects MISGAM and ENIGMA. We would like to thank Ettore Aldrovandi, Lev Borisov, Jean-Louis Colliot-The´le`ne, Andrew Kresch, Fabio Perroni, Ilya Tyomkin and Angelo Vistoli for helpful discussions; in particular Aldrovandi for explanations about group-stacks and reference [11], Borisov for pointing out a mistake in a preliminary version, Colliot-The´le`ne for [20], §6, Tyomkin for [9] and Vistoli for useful information about the classiﬁcation of gerbes.

1. Notations and background

1.1. Conventions and notations. A scheme will be a separated scheme of ﬁnite type over C. A variety will be a reduced, irreducible scheme. A point will be a C-valued point. The smooth locus of a variety X will be denoted by Xsm.

We work in the e´tale topology. For an algebraic stack X, we will write that x is a point of X or just x A X to mean that x is an object in X C ; we denote by Aut x the automorphism group of the point x. We will say that a morphism between stacks is unique if it is unique up to a unique 2-arrow. As usual, we denote Gm the sheaf of invertible sections in OX on the e´tale site of X.

1.2. Smooth Deligne-Mumford stacks and orbifolds. A Deligne-Mumford stack will be a separated Deligne-Mumford stack of ﬁnite type over C; we will always assume that its coarse moduli space is a scheme. An orbifold will be a smooth Deligne-Mumford stack with trivial generic stabilizer. For a smooth Deligne-Mumford stack X, we denote by eX or just e the natural morphism from X to its coarse moduli space X, which is a variety with ﬁnite quotient singularities.

Let i : U X be an open embedding of irreducible smooth Deligne-Mumford stacks with complement! of codimension at least 2. We have that:

The natural map i : Pic X Pic U is an isomorphism. ! For any line bundle L A Pic X , the natural morphism i : H 0 X; L H 0 U; i L is also an isomorphism. !

The inertia stack, denoted by I X , is defined to be the fibered product I X : X X X X. A point of I X is a pair x; g with x A X and g A Aut x . The inertia stack of a smooth Deligne-Mumford stack is smooth but di¤erent components will in general have di¤erent dimensions. The natural morphism I X X is representable, unramified, proper and a relative group scheme. The identity section ! gives an irreducible component canonically isomorphic to X; all other components are called twisted sectors.

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 205

A smooth Deligne-Mumford stack of dimension d is an orbifold if and only if all the twisted sectors have dimension e d 1, and is canonical if and only if all twisted sectors have dimension e d 2. Remark 1.1 (sheaves on global quotients). According to [33], Appendix, a coherent sheaf on a Deligne-Mumford stack Z=G is a G-equivariant sheaf on Z, i.e., the data of a coherent sheaf LZ on Z and for every g A G an isomorphism jg : LZ gLZ such that j h j j . ! gh g h Notice that if Z is a subvariety of Cn of codimension higher or equal than two then an invertible sheaf on Z=G is the structure sheaf OZ and a one dimensional representation of G, i.e., w : G C. A global section of such an invertible sheaf on Z=G is a w- ! equivariant global section of OZ.

We end this subsection with a proposition extending to stacks a property of separated schemes. We will prove it in Appendix A.

Proposition 1.2. Let X and Y be two Deligne-Mumford stacks. Assume that X is normal and Y is separated. Let i : U , X be a dominant open immersion of the Deligne- Mumford stack U.IfF; G : X Y are! two morphisms of stacks such that there exits a b ! 2-arrow F i G i then there exists a unique 2-arrow a : F G such that a idi b. ) ) The previous proposition is well known for X a reduced scheme and Y a separated scheme. Nevertheless, if X is not a normal stack we have the following counter-example: Y B X X Y Let be m2. Let be a rational curve with one node. Let F1 : (resp. F2) be a stack morphism given by a non-trivial (resp. trivial) double cover of X.! Putting U X node , the proposition is false. nf g

1.3. Root constructions. For this subsection we refer to the paper of Cadman [12] (see also [2], Appendix B). In this part X will be a Deligne-Mumford stack over C (it is enough to assume that X is Artin.)

1.3.a. Root of an invertible sheaf. This part follows closely [2], Appendix B. Let L be an invertible sheaf on the Deligne-Mumford stack X. Let b be a positive integer. We denote by b L=X the following fiber product pffiffiffiffiffiffiffiffiffiffi b L=X BC ! pffiffiffiffiffiffiffiffiffiffi r 5b ? ? ? L ? X? B?C y ! y nb where 5b : BC BC sends an invertible sheaf M over a scheme S to M . More explicitly, an object! of b L=X over f : S X is a couple M; j where M is an invertible nb @! sheaf M on the schemepffiffiffiffiffiffiffiffiffiffiS and j : M f L is an isomorphism. The arrows are defined in an obvious way. !

b 1=b The morphism L=X BC corresponds to an invertible sheaf, denoted by L in b ! [8], on L=X whosepb-thffiffiffiffiffiffiffiffiffiffi power is isomorphic to the pullback of L. pffiffiffiffiffiffiffiffiffiffi

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206 Fantechi, Mann and Nironi, Deligne-Mumford stacks

b X X The stack L= is a mb-banded gerbe over (see the second paragraph of Subsec- tion 6.1 below).p Theffiffiffiffiffiffiffiffiffiffi Kummer exact sequence

5b 1 m Gm Gm 1 ! b ! ! ! 1 X G 2 X induces the boundary morphism q : Het´ ; m Het´ ; mb . The cohomology class of the b 2 ! 1 m -banded gerbe L=X in H X; m is the image by q of the class L A H X; Gm . b et´ b et´ pffiffiffiffiffiffiffiffiffiffi The gerbe is trivial if and only if the invertible sheaf L has a b-th root in Pic X . More b X b X generally, the gerbe L= is isomorphic, as a mb-banded gerbe, to L0= if and only if L L in Pic X =pb PicffiffiffiffiffiffiffiffiffiffiX . We have the following morphism of shortpffiffiffiffiffiffiffiffiffiffiffiffi exact sequences: 0 b 0 Z Z Z=bZ 0 ! ! ! ! 1:3 ? ? ? ? 0 Pic?X Pic b?L=X Z=bZ 0 ! y ! y ! ! pffiffiffiffiffiffiffiffiffiffi where the first and second vertical morphisms are defined by 1 L and 1 L1=b, respectively. 7! 7!

1.3.b. Roots of e¤ective Cartier divisors. In the articles [12] and [2], the authors de- ﬁne the notion of root of an invertible sheaf with a section on an algebraic stack: here, we only consider roots of e¤ective Cartier divisors on a smooth algebraic stack, since this is what we will use.

n n Let n be a positive integer. Consider the quotient stack A = C where the action n n n of C is given multiplication coordinates by coordinates. Notice that A = C is the n moduli stack of n line bundles with n global sections. Let a : a1; ...; an A N>0 be an n n n n n-tuple. Denote by 5a : A = C A = C the stack morphism deﬁned by sending ai ai ! n n xi x and li l where xi (resp. li) are coordinates of A (resp. C ). 7! i 7! i

Let X be a smooth algebraic stack. Let D : D1; ...; Dn be n e¤ective Cartier divisors. The a-th root of X; D is the fiber product a n n D=X A = C ! pffiffiffiffiffiffiffiffiffiffiffi p r 5a ? ? ? D n ? n X? A =?C : y ! y a n n The morphism D=X A = C corresponds to the e¤ective Cartier divisors DD~ ~ ~ !~ 1 : DD1; ...; DDn p, whereffiffiffiffiffiffiffiffiffiffiffi DDi is the reduced closed substack p Di red. More explicitly, an a ~ ~ object of D=X over a scheme S is a couple f ; DD1; ...; DDn where f : S X is a mor- ~ ! phism andp forffiffiffiffiffiffiffiffiffiffiffi any i, Di is an e¤ective divisor on S such that aiDDi f Di. We have the following properties:

a ai (1) The fiber product of Di=X over X is isomorphic to D=X (cf. [12], Remark 2.2.5). pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 207

a (2) The canonical morphism D=X X is an isomorphism over X Di. ! n pffiffiffiffiffiffiffiffiffiffiffi Si

(3) If X is smooth, each Di is smooth and the Di have simple normal crossing then a D=X is smooth (cf. Section 2.1 of [8]) and DD~i have simple normal crossing. pffiffiffiffiffiffiffiffiffiffiffi (4) We have the following morphism of short exact sequences (cf. [12], Corollary 3.1.2)

n n a n 0 Z Z Z=aiZ 0 ! ! ! i 1 ! Q 1:4 ? ? ? ? n ?X p a?D X q 0 Picy Pic y = Z=aiZ 0 ! ! ! i 1 ! pffiffiffiffiffiffiffiffiffiffiffi Q where the first and second vertical morphisms are defined by ei O Di and ei O DD~i , respectively. Every invertible sheaf L A Pic a D=X can be written7! in a unique7! way as n ~ pffiffiffiffiffiffiffiffiffiffiffi L G pM n O kiDDi where M A Pic X and 0 e ki < ai; the morphism q maps L to i 1 k1; ...; kn . Q

We finish this section with the following observation. Let D1 and D2 be two e¤ective a a; a Cartier divisors on X such that D X D 3j. The stacks D W D =X and D ; D =X 1 2 1 2 1 2 are not isomorphic. Indeed, the stabilizer group at any pointpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the preimage ofpxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA D1 X D2 a a; a in D W D =X (resp. D ; D =X)ism (resp. m m ). 1 2 1 2 a a a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1.4. Rigidification. In this subsection, we sum up some results on the rigidification of an irreducible d-dimensional smooth Deligne-Mumford stack X. Intuitively, the rigidification of X by a central subgroup G of the generic stabilizer is constructed as follows: first, one constructs a prestack where the objects are the same and the automorphism groups of each object x are the quotient AutX x =G; then the rigidification X( G is the stackification of this prestack. For the most general construction we refer to [3], Appendix A (see also [1], Section 5.1, [32] and [2], Appendix C).

We consider the union I gen X H I X of all d-dimensional components of I X ;itis a subsheaf of groups of I X over X which is called the generic stabilizer. Most of the time in this article, we will rigidify by the generic stabilizer. In this case, we write Xrig in order to mean X( I gen X and call it the rigidiﬁcation. The rigidiﬁcation r : X Xrig has the following properties: ! (1) The coarse moduli space of Xrig is the coarse moduli space of X.

(2) Xrig is an orbifold.

(3) If X is an orbifold then Xrig is X.

(4) The morphism r makes X into a gerbe over Xrig.

We refer to [1], Theorem 5.1.5(2), for the proof of the following proposition.

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208 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Proposition 1.5 (universal property of the rigidiﬁcation). Let X be a smooth Deligne- Mumford stack. Let Y be an orbifold. Let f : X Y be a dominant stack morphism. Then there exists g : Xrig Y and a 2-morphism a :!g r f such that the following is 2-

commutative: ! )

X r Xrig !

bg f ? ? Y?: y If there exists g : Xrig Y and a 2-morphism a : g r f satisfying the same property 0 ! 0 0 ) then there exists a unique g : g g such that a g idr a . 0 ) 0 1.5. Diagonalizable group schemes. In this short subsection, we recall some results on diagonalizable groups.

Deﬁnition 1.6. A group scheme G over Spec C will be called diagonalizable if it is isomorphic to the product of a torus and a ﬁnite abelian group.

We use multiplicative notation for diagonalizable group. For any diagonalizable 4 group G, its character group G : Hom G; C is a ﬁnitely generated abelian group (or coherent Z-module). The duality contravariant functor G G4 induces an equivalence of categories from diagonalizable to coherent Z-module. Its7! inverse functor is given by 4 F GF : Hom F; C . Both G G and F GF are contravariant and exact. 7! 7! 7! 1.6. Toric varieties. We recall some results on toric varieties that can be found in [17] (see also [15]). The principal construction used in this paper is the description of toric varieties as global quotients found by Cox (see [13]).

We ﬁx a torus T, and denote by M T4 the lattice of characters and by N : Hom M; Z the lattice of one-parameter subgroups. A toric variety X with torus T corresponds to a fan S X , or just S,inNQ : N nZ Q, which we will always assume to be simplicial.

S Let r1; ...; rn be the one-dimensional cones, called rays, of . For any ray ri, denote by vi the unique generator of r X N. For any i in 1; ...; n , we denote by Di the irreduc- i f g ible T-invariant Weil divisor deﬁned by the ray ri. The free abelian group of T-invariant Weil divisor is denoted by L. n Let i : M L be the morphism that sends m m vi . If the rays span NQ (which ! 7! i 1 P is not a strong assumption2)), the morphism i is injective, and ﬁts into an exact sequence in Coh Z i 1:7 0 M L A 0; ! ! ! !

2) Indeed, if the rays do not span NQ then X is isomorphic to the product of a torus and a toric variety XX~ whose rays span NN~Q.

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 209 where A is the class group of X (i.e., the Chow group A1 X ). We deduce that the short exact sequence of diagonalizable groups

1:8 1 GA GL T 1: ! ! ! ! n n Let ZS H C be the GL C -invariant open subset deﬁned as ZS : Zs, where S sSA Zs : x xi 3 0ifri B s . The induced action of GA on ZS has ﬁnite stabilizers (by the f j g n simpliciality assumption) and X is the geometric quotient ZS=GA, with torus C =GA (see [13], Theorem 2.1). For any i A 1; ...; n , the T-invariant Weil divisor Di H X is the geometric quotient f g

1:9 xi 0 X ZS =GA: f g

If X is smooth then the natural morphism L Pic X given by ei OX Di is surjective and has kernel M; in other words, it induces! a natural isomorphism7!A Pic X . ! If X is a d-dimensional toric variety, we will write X 0 for the union of the orbits of dimension f d 1; in other words, X 0 is the toric variety associated to the fan 0 Se1 : s A S dim s e 1 . The toric variety X is always smooth and the toric divisors 0 f j g Dr are smooth, disjoint, and homogeneous under the T-action (with stabilizer the one- dimensional subgroup which is the image of r).

1.7. Picard stacks and action of a Picard stack. Deligne deﬁned Picard stacks in [7], Expose´ XVIII, as stacks analogous to sheaves of abelian groups. For the readers convenience, we collect here a sketch of the deﬁnition and the main properties we need; details can be found in [7], Expose´ XVIII, and also in [26], Section 14.

Here we summarize the deﬁnition of a Picard stack. For the details we refer to Deﬁ- nition B.1.

Deﬁnition 1.10. Let G be a stack over a base scheme S.APicard stack G over S is given by the following set of data:

a multiplication stack morphism m : G G G, also denoted by ! m g ; g g g ; 1 2 1 2 an associativity 2-arrow g g g g g g ; 1 2 3 ) 1 2 3 a commutativity 2-arrow g g g g . 1 2 ) 2 1 These data satisfy some compatibility relations, which we list in B.1.

The deﬁnition implies that there also exists an identity e : S G and an inverse i : G G with the obvious properties; in particular, a 2-arrow e : e !g g. ! )

Deﬁnition 1.11 (see [7], Section 1.4.6). Let G, G0 be two Picard stacks. A morphism of Picard stacks F : G G0 is a morphism of stacks and a 2-arrow a such that for any two ! objects g1, g2 in G, we have a F g g F g F g : 1 2) 1 2

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210 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Again we refer to Appendix B for the list of compatibilities satisﬁed by a. The Picard stacks over S form a category where the objects are Picard stacks and morphisms are equivalence classes of morphisms of Picard stacks.

Remark 1.12. To any complex G : G 1 G0 of sheaves of abelian groups, we ! can associate a Picard stack G. In this paper, G will be a complex of diagonalizable groups 1 0 and the associated Picard stack is the quotient stack G =G . Proposition 1.13 (see [7], Proposition 1.4.15). The functor that associates to a length 1 complex of sheaves of abelian groups a Picard stack induces an equivalence of categories 1; 0 between the derived category, denoted by D S; Z , of length 1 complexes of sheaves of abelian groups and the category of Picard stacks.

In particular, if G is any sheaf of abelian groups on the base scheme S, the quotient S=G , i.e. the gerbe BG, is naturally a Picard stack. We finish this section with a sketch of the definition of an action of a Picard stack on a stack. This is a generalization of the action of a group scheme on a stack defined by Romagny in [32]. We refer to Definition B.12 for the details.

Deﬁnition 1.14 (action of a Picard stack). Let G be a Picard stack. Denote by e the neutral section and by the corresponding 2-arrow. Let X be a stack. An action of G on X is the following data:

a stack morphism a : G X X, also denoted by a g; x g x; ! a 2-arrow e x x; ) an associativity 2-arrow g g x g g x . 1 2 ) 1 2 These data satisfy some compatibility relations, which we list in Appendix B.

2. Deligne-Mumford tori

In this section we deﬁne Deligne-Mumford tori which will play the role of the torus for a toric variety.

We start with a technical lemma.

Lemma 2.1. Let f : A0 A1 be a morphism of ﬁnitely generated abelian groups such that ker f is free. In the derived! category of complexes of ﬁnitely generated abelian groups 0 of length 1, the complex A0 A1 is isomorphic to ker f coker f . ! ! Proof. We have a morphism of complexes

f ff~ A0 A1 A0=A0 A1=A0 ! ! tor ! tor induced by the quotient morphisms. As ker f is free, we deduce after a diagram chasing that this morphism is a quasi-isomorphism of complexes. In the derived category, we replace

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 211

1 0 Zl Q Zd l A =Ator with a projective resolution . Then the mapping cone of the morphism 0 0 Zl !Zd l ~ 0 0 Zl Zd l of complexes 0 A =Ator Q : is ffQ : A =Ator which is ! !ff~ ! ! quasi-isomorphic to A0=A0 A1=A0 . A morphism of free abelian groups f is quasi- tor 0 tor isomorphic to the complex ker!f coker f and this ﬁnishes the proof. r ! The reader who is familiar with the article [10] has probably recognized part of the construction of the stack associated to a stacky fan.

Remark 2.2. Let f : A0 A1 be a morphism of ﬁnitely generated abelian groups as ! in the above lemma. Applying the contravariant functor Hom ; C of Section 1.5 to the G 0 1 f complex A A , we get a length 1 complex of diagonalizable groups G 1 G 0 . Ac- ! A ! A cording to Remark 1.12, the associated Picard stack GA0 =GA1 is a Deligne-Mumford stack if and only if the cokernel of f is ﬁnite.

n 1 Example 2.3. Let w0; ...; wn be in N>0. Let f : Z Z that sends a0; ...; an to n ! wiai. We have that ker f Z and coker f Z=dZ where d : gcd w0; ...; wn . Hence, n B Pthe associated Picard stack is C md . Definition 2.4. A Deligne-Mumford torus is a Picard stack over Spec C which is 0 1 obtained as a quotient GA0 =GA1 , where f : A A is a morphism of finitely generated abelian groups such that ker f is free and coker f!is finite.

Let G be a finite abelian group. Notice that BG is a Deligne-Mumford torus. Recall that by Proposition 1.13, T BG has a natural structure of Picard stack. Definition 2.5. A short exact sequence of Picard S-stacks is the sequence of morphisms of Picard S-stacks associated to a distinguished triangle in D 1; 0 S . Proposition 2.6. Any Deligne-Mumford torus T is isomorphic as Picard stack to T BG where T is a torus and G is a finite abelian group. 0 1 Proof. Let T G 0 =G 1 with f : A A as above. The distinguished triangle G A A f ! 1; 0 ker Gf 0 G 1 G 0 0 coker Gf in the derived category D Spec C in- ! ! A ! A ! ! duces an exact sequence of Picard stacks 1 BG T T 1 where T : GA0 =GA1 . Proposition 1.13 and Lemma 2.1 imply that there! is a! non-canonical! ! isomorphism of Picard stacks T 1 BG T. r Note that the scheme T in the previous proof is the coarse moduli space of T.

3. Deﬁnition of toric Deligne-Mumford stacks

Deﬁnition 3.1. A smooth toric Deligne-Mumford stack is a smooth separated Deligne-Mumford stack X together with an open immersion of a Deligne-Mumford torus i : T , X with dense image such that the action of T on itself extends to an action a : T !X X. ! As in this paper all toric Deligne-Mumford stacks are smooth, we will write toric Deligne-Mumford stack instead of smooth toric Deligne-Mumford stack. We will see later

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212 Fantechi, Mann and Nironi, Deligne-Mumford stacks in Theorem 7.24 that our definition a posteriori coincides with that in [10] via stacky fans. It seems natural to define a toric Deligne-Mumford stack by replacing smooth with normal in the above definition. All the definitions and results in this section apply also in this case, with the exception of Proposition 3.6 and Lemma 3.8. Ilya Tyomkin is currently working on this. A toric orbifold is a toric Deligne-Mumford stack with generically trivial stabilizer. A toric Deligne-Mumford stack is a toric orbifold if and only if its Deligne-Mumford torus is an ordinary torus. Hence, the notion of toric orbifold is the same as the one used in [22], Theorem 1.3.

Remark 3.2. (1) Separatedness of X and Proposition 1.2 imply that the action of T on X is uniquely determined by i.

(2) Notice that we have assumed in Section 1.2 that the coarse moduli space is a scheme. Without this assumption, if the coarse moduli space X of a toric Deligne-Mumford stack is a smooth and complete algebraic space then the main theorem of Bialynicki-Birula in [9] implies that X is a scheme. We dont know whether such an assumption is necessary in general.

(3) A toric variety admits a structure of toric Deligne-Mumford stack if and only if it is smooth.

Proposition 3.3. Let X be a smooth Deligne-Mumford stack together with an open dense immersion of a Deligne-Mumford torus i : T , X such that the action of T on itself extends to a stack morphism a : T X X. Then! the stack morphism a induces naturally an action of T on X. !

Proof. We will deﬁne a 2-arrow h : a e; idX idX and a 2-arrow )

s : a m; idX a idX; a ) such that they verify conditions (1) and (2) of Deﬁnition B.12. We will only prove the existence of h because the existence of s and the relations (1) and (2) follow with a similar argument.

Denote by e : Spec C T the neutral element of T and by m : T T T the ! ! multiplication on T. Denote by e the 2-arrow m e; idT idT. As the stack mor- ) phism a extends m, we have a 2-arrow a : a idT; i i m. Denote by b the 2-arrow ) e; idX i idT; i e; idT . Consider the two stack morphisms: )

idX TXXi :

a e; idX Applying Proposition 1.2 with the composition of the following 2-arrows

id b a id e; idT id e a i a e; idX i a idT; i e; idT i m e; idT i idT idX i; ) ) ) we deduce the existence of h : a e; idX idX. r )

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 213

Definition 3.4. Let X (resp. X0) be a toric Deligne-Mumford stack with Deligne- Mumford torus T (resp. T0). A morphism of toric Deligne-Mumford stacks F : X X0 is ! a morphism of stacks between X and X0 which extends a morphism of Deligne-Mumford tori T T0: ! Remark 3.5. The extended morphism F in the previous definition is unique by Pro- position 1.2. Moreover the definition of morphism between Picard stacks and Proposition 1.2 provide us the following 2-cartesian diagram:

F; F T X T j X0 T0 ! a r a ? ? 0 ? F ? X? X?0: y ! y Proposition 3.6. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T. Let X (resp. T) be the coarse moduli space of X (resp. T). Then X has a structure of simplicial toric variety with torus T where the open dense immersion i : T , X and the action a : T X X is induced respectively by i : T , X and a : T X !X. ! ! ! Proof. The morphisms i and a induce morphisms on the coarse moduli spaces i : T X and a : T X X, by the universal property of the coarse moduli space. It is immediate! to verify that i!is an open embedding with dense image and a is an action, extending the action of T on itself. On the other hand, since X is the coarse moduli space of X, it is a normal separated variety with ﬁnite quotient singularities. Therefore X is a toric variety, and it is simplicial by [21], §7.6, p. 121 (see also [15], Theorem 3.1, p. 28). r

Remark 3.7 (divisor multiplicities). According to [26], Corollary 5.6.1, the structure morphism e : X X induces a bijection on reduced closed substacks. For each ! 1 i 1; ...; n, denote by Di H X the reduced closed substack with support e Di . Since Di X Xsm is a Cartier divisor, there exists a unique positive integer ai such that 1 1 e Di X X ai Di X e X . We call a a ; ...; an the divisor multiplicities of X. sm sm 1 Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T T BG. By Appendix B, we have that BG acts on X. Proposition B.15 implies that we have an e´tale morphism j : G X I gen X . ! Lemma 3.8. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T T BG. The morphism j : G X I gen X is an isomorphism. ! Proof. As the stack X is separated, we have that the natural morphism I X X is proper. As the projection G X X is a proper morphism, the morphism jis! also a proper morphism. Its image contains ! the substack I T I gen T which is open and dense in I gen X . We deduce that the morphism j is birational. As the morphism j is e´tale, it is quasi-finite (cf. [19], Expose´ I, §3). The morphism j is proper hence closed and as its image contains the open dense torus, j is surjective. The morphism j is a representable, birational, surjective and quasi-finite morphism to the smooth Deligne-Mumford stack X. The stacky Zariskis main theorem C.1 finishes the proof. r

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214 Fantechi, Mann and Nironi, Deligne-Mumford stacks

4. Canonical toric Deligne-Mumford stacks

In §4.1 we define the canonical smooth Deligne-Mumford stack associated to a variety with finite quotient singularities and we show that a canonical smooth Deligne- Mumford stack satisfies a universal property (Theorem 4.6). This should be well known, but we include it for the readers convenience.

In §4.2, we characterize the canonical toric Deligne-Mumford stack via its coarse moduli space.

4.1. Canonical smooth Deligne-Mumford stacks. In this subsection, we do not assume that smooth Deligne-Mumford stacks are toric. First, we deﬁne canonical smooth Deligne-Mumford stacks and then we prove their universal property.

We recall a classical result.

Lemma 4.1. Let S be a smooth variety, and T be an a‰ne scheme. Let S 0 H Sbean open subvariety such that the complement has codimension at least 2 in S. Let f : S 0 Tbe a morphism. Then the morphism f extends uniquely to a morphism S T. ! ! Proof. The morphism f corresponds to an algebra homomorphism

K T G S 0; OS : ! 0 G O G O Since the complement has codimension 2, the restriction map S; S S 0; S 0 is an isomorphism. r !

Deﬁnition 4.2. (1) A dominant morphism f : V W of irreducible varieties is called codimension preserving if, for any irreducible closed! subvariety Z of W and every 1 irreducible component ZV of f Z , one has codimV ZV codimW Z. (2) A dominant morphism of orbifolds is called codimension preserving if the induced morphism on every irreducible component of the coarse moduli spaces is codimension preserving.

Remark 4.3. For any Deligne-Mumford stack, the structure morphism to the coarse moduli space is codimension preserving. Every ﬂat morphism and in particular every smooth and e´tale morphism is codimension preserving. A composition of codimension preserving morphisms is codimension preserving.

Deﬁnition 4.4. Let X be an irreducible d-dimensional smooth Deligne-Mumford stack. Let e : X X be the structure morphism to the coarse moduli space. The Deligne- Mumford stack !X will be called canonical if the locus where e is not an isomorphism has dimension e d 2. Remark 4.5. Let X be a smooth canonical stack

(1) The locus where the structure map to the coarse moduli space e : X X is an isomorphism is precisely e 1 X , where X is the smooth locus of X. ! sm sm

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 215

(2) The composition of the following isomorphisms

1 F 1 F F 1 F A X A X Pic X Pic e X Pic X ! sm! sm! sm ! 1 is the map sending D to O e D . Theorem 4.6 (universal property of canonical smooth Deligne-Mumford stacks). Let Y be a canonical smooth Deligne-Mumford stack, e : Y Y the morphism to the coarse moduli space, and f : X Y a dominant codimension preserving! morphism with X an orbifold. Then there exists a! unique, up to a unique 2-arrow, g : X Y such that the following

diagram is commutative: !

X b !g Y

! e f ? ? Y?: y Proof. We first prove uniqueness. Any two morphisms g, g making the diagram 1 1 commute must agree on the open dense subscheme f Ysm . Put i : f Ysm , X. Since Y is assumed to be separated, by Proposition 1.2, there exists a unique a: g !g such that ! a idi id. By uniqueness, it is enough to prove the result e´tale locally in Y, so we can assume that Y V=G where V is a smooth a‰ne variety and G a finite group acting on V without pseudo-reflections. It is enough to show that there exists an e´tale surjective morphism p : U X with U a smooth variety and a morphism g : U Y such that f p e g. In fact,! g is defined from g by descent, with the appropriate! compatibility conditions being taken care of by the uniqueness part. In this case Y V=G, and Y0 : V0=G where 1 V H V is the open locus where G acts freely. Let U : f p Y .As V =G is 0 0 0 0 isomorphic to Y0, we have a natural morphism U0 V0=G . This morphism defines a principal G-bundle P on U and a G-equivariant morphism! s : P V .

0 0 0 0 ! 0 s

P 0 V

0 ! 0

s PV !

U0 V0=G !

4:7 U V=G

P; s

f p Y0 V0=G e

Y V= G

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216 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Since the U U0 has codimension f 2, the principal G-bundle P0 extends uniquely to a principal G-bundlen P over U, and by Lemma 4.1 (since V is a‰ne) the G-equivariant morphism s0 : P0 V0 extends to a morphism s : P V which is again G-equivariant, yielding a morphism! g : U V=G . The construction! above is summarized in the 2- commutative diagram (4.7) where! the squares are 2-cartesian. r

Corollary 4.8. Let X (resp. Y) be a canonical smooth Deligne-Mumford stack with coarse moduli space X (resp. Y). Let f : X Y be an isomorphism. Then there is a unique isomorphism f : X Y inducing f . ! ! Proof. It is enough to apply the theorem twice, reversing the role of X and Y. r

Remark 4.9. One can use the corollary to prove the classical fact that every variety Y with finite quotient singularities is the coarse moduli space of a canonical smooth Deligne-Mumford stack unique up to rigid isomorphism, which we denote by Ycan (do it e´tale locally and then glue). If Y is the geometric quotient Z=G where Z is a smooth variety and G is a group without pseudo-reflections acting with finite stabilizers, then Ycan Z=G . Notice that this is the case of simplicial toric varieties (cf. Section 1.6). We finish this section with a corollary that will play an important role.

Corollary 4.10. Let X be a smooth Deligne-Mumford stack with coarse moduli space e : X X. There is a unique morphism X Xcan through which e factors. ! ! Proof. Apply the theorem with Y X, Y Xcan and f e. r 4.2. The canonical stack of a simplicial toric variety. In this section, we study the canonical stack associated to a simplicial toric variety.

The main result of this section is the following theorem.

Theorem 4.11. Let X be a simplicial toric variety with torus T. Its canonical stack Xcan has a natural structure of toric orbifold such that the action acan : T Xcan Xcan lifts the action a : T X X. ! !

Proof. Denote by S the fan in N nZ Q of the toric variety X. Without loss of generality, we can assume that the rays of S generate N nZ Q, so that X ZS=GA (cf. §1.6). The subvariety of points where GA acts with non-trivial stabilizers has codimension f 2. can Remark 4.9 implies that the canonical stack X is isomorphic to ZS=GA . Let n can n T : C =GA be the torus of the toric variety X. Notice that T C =GA is open dense and isomorphic via e T can to T. Proposition 3.3 and the universal property (see The- orem 4.6) of the canonical stackj imply that the action of T on X lifts to an action of T on Xcan. r

Remark 4.12. (1) Under the hypothesis of Theorem 4.11, we have that the restriction of the structure morphism e : Xcan X to Tcan is an isomorphism with T. ! (2) Let X be a canonical toric Deligne-Mumford stack with Deligne-Mumford torus T T with coarse moduli space the simplicial toric variety X. Denote by

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 217

S H NQ : N n Q the fan of X. Assume that the rays of S generate NQ. The proof above 1 shows that X ZS=GA where GA Hom A X ; C Hom Pic X ; C (cf. Remark 4.5(2)).

Corollary 4.13. Let X be a canonical toric Deligne-Mumford stack with torus T T and coarse moduli space the simplicial toric variety X. Denote S H NQ the fan of X.

(1) The boundary divisor X T is a simple normal crossing divisor, with irreducible n components, denoted by Di. Moreover, if the rays of S generate NQ, then the divisor Di is isomorphic to Zi=GA where Zi xi 0 X ZS. f g 1 e (2) The composition morphism L A X Pic X sends ei to OX Di . ! ! Proof. The ﬁrst point of the corollary follows from the fact that the inverse image n n inside ZS of the torus T C =GA is C . The second part of the corollary follows from the exact sequence (1.7) and Remark 4.5(2). r

Remark 4.14. Let X be a canonical toric Deligne-Mumford stack with coarse moduli space X.

(1) Denote by S the fan of X in NQ. If the rays of S span NQ, from the corollary and the exact sequence (1.7), we have the exact sequence

0 M L Pic X 0: ! ! ! !

(2) For any i A 1; ...; n , the divisor Di is Cartier. Hence it corresponds to the inver- f g tible sheaf O Di with the canonical section si. Using Remark 1.1, the invertible sheaf n pi O Di is associated to the representation GA GL C C where pi is the i-th pro- ! ! jection. Moreover, the canonical section si is the i-th coordinate of ZS.

(3) Let X be a canonical toric Deligne-Mumford stack, then all divisor multiplicities of X are equal to 1 (for the deﬁnition of divisor multiplicity see Remark 3.7).

5. Toric orbifolds

In this section, we only consider toric Deligne-Mumford stacks with trivial generic stabilizer that is toric orbifolds.

Let X be a smooth Deligne-Mumford stack with coarse moduli space X. By Proposi- tion 3.6 and Theorem 4.11, the canonical stack Xcan has an induced structure of toric can orbifold. Denote by eX : X X (resp. eX can : X X) the morphism to the coarse moduli space. Theorem 4.6! implies that there exists! a unique f : X Xcan such that ! eX can f eX. Proposition 5.1. Let X be a toric orbifold with torus T and coarse moduli space X. The canonical morphism f : X Xcan is a morphism of toric Deligne-Mumford stacks where Xcan is endowed with the induced! structure of toric orbifold.

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218 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Proof. The universal property of the canonical stack (cf. Theorem 4.6) applied to id : T T implies that f : T Tcan. r ! jT !

Tcan Notice that the morphism f T : T in the proof above is an isomorphism because X is a toric orbifold. j !

Denote Dcan : Dcan; ...; Dcan (cf. Section 1.3.b). 1 n

Theorem 5.2. (1) Let X be a simplicial toric variety with torus T. Denote by S a fan S a n of X. For each ray ri of , choose ai in N>0. Denote : a1; ...; an A N>0 . Then a Dcan=Xcan has a unique structure of toric orbifold with torus T such that the canonical a pmorphismffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip : Dcan=Xcan Xcan is a morphism of toric Deligne-Mumford stacks with ! divisor multiplicitiespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia.

(2) Let X be a toric orbifold with coarse moduli space X. Let a : a1; ...; an be its divisors multiplicities. Then X is naturally isomorphic as toric Deligne-Mumford stack to a Dcan=Xcan defined in (1). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Proof. (1) Let Tcan L Xcan be the inverse image of T (which is isomorphic to T). 1 a can can can Note that p T L D =X is isomorphic to T by property (2) of Section 1.3.b. a Let j : T Dcan=Xpcanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibe the dominant open embedding. We need to prove that T a! acts on Dpcanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=Xcan compatibly with j. We know that T acts on Xcan. To define T a Dcanpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=Xcan a Dcan=Xcan we use the universal property and the fact that can ! D pLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXcan is T-invariant.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

can D X (2) For any i A 1; ...; n , denote by Di, Di , i the divisor corresponding to the f Xcan g X ray ri in respectively X, and . Theorem 4.11 implies there exists a unique morphism can f : X X such that eX can f eX. By deﬁnition of the divisors multiplicities, for any ! 1 can D X D X D X D X ray ri, we have f Di ai i . The Cartier divisors : 1 ; ...; n de- n n ﬁne a morphism X A = C such that the following diagram is 2-commutative: !

D X n n X A = C ! 5:3 f 5a ? ? D can ?can n ? n X? A =?C y ! y where the morphism 5a is deﬁned in Section 1.3.b. By the universal property of ﬁber product, we deduce a unique morphism g : X a Dcan=Xcan such that the following diagram is

! strictly commutative: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g a can can

X D =X ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p f ? ? X?can y :

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 219

We will use the Zariskis main theorem (cf. Theorem C.1) to prove that g is an isomorphism. We first notice that a Dcan=Xcan is smooth for property (3) in §1.3.b. As, the Xcan can can restriction of g over DipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX Dj is an isomorphism, the morphism g is birational. i; j can can S Notice that Di X Dj is a subset of codimension f 2. The morphism g is proper, Si; j hence closed, so we deduce that g is also surjective because its image contains the dense torus. Let us show that g is representable and e´tale. Let S be a scheme. Consider the following 2-cartesian diagram:

g Y S ! r ? ? ? g ? X? a Dcan?=Xcan: y ! y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let U Y be an e´tale atlas of Y. First we observe that the morphism U S, denote it by gg~, must! be flat, so that the morphism g is flat too. To verify this we can! apply [27], Thm. 23.1, using that both S and U are smooth and the dimension of the fibers of gg~ is constantly zero. To prove that the dimension of the fibers is zero we just need to observe that both p and f are quasi-finite, since they are morphisms from a stack to its coarse moduli space, and f factors through g so that it must be quasi-finite too. We now note that the morphism U S is e´tale away from a codimension f 2 subset, so we can apply the theorem of purity! of branch locus (cf. [5], Theorem 6.8, p. 125) and deduce that U S is e´tale, i.e., g : Y S is e´tale. Without loss of generality we can assume that S is actually! an atlas; we assume! that Y is a stack and we prove that it must be actually a scheme. First of all we observe that it cannot have generically non-trivial stabilizer, since the morphism Y X is representable it must induce an injection of the stabilizer at each geometric point [4],! but X is an orbifold so that Y must be an orbifold too. There exists an e´tale representable map V=K Y where V is a smooth variety and K is a finite group. Hence the induced map V S!is e´tale. By the universal property, it factors via the coarse moduli space V=K, and the! map V V=K is not injective on tangent vectors unless K is acting freely, hence V V=K cannot! be e´tale unless Y has trivial stabilizers everywhere. We now observe that! the morphism V=K S is still birational surjective and quasi-finite, using Zariskis main theorem for schemes! we can deduce that it is an isomorphism, in particular it is e´tale and this implies that V V=K must be e´tale. We conclude that Y is a scheme, i.e., g is representable and e´tale.! So it is also quasi-finite (cf. [19], Expose´ I, §3).

As the morphism g is representable, surjective, birational and quasi-ﬁnite, the stacky Zariskis main theorem C.1 implies that g is an isomorphism. r

The following corollary is a consequence of property (3) of Section 1.3.b and Theo- rem 5.2.

Corollary 5.4. Let X be a toric orbifold with coarse moduli space X. The reduced closed substack X T is a simple normal crossing divisor. n Remark 5.5. Let X be a toric orbifold with coarse moduli space X. Diagram (1.4) and Theorem 5.2 imply that we have the following morphism of exact sequences:

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220 Fantechi, Mann and Nironi, Deligne-Mumford stacks

n n a n 0 Z Z Z=aiZ 0 ! ! ! i 1 ! L 5:6 ? ? ? ? n X? can f ?X q 0 Pic y Picy Z=aiZ 0 ! ! ! i 1 ! L can where the vertical morphisms send 1 O D and 1 O Di . 7! i 7!

6. Toric Deligne-Mumford stacks

In this section we will show that each toric Deligne-Mumford stack is isomorphic to a fibered product of root stacks on its rigidification. To prove this theorem, we will recall in Section 6.1 the relation between banded gerbes and root constructions. Then we will show in Theorem 6.11 that any toric Deligne-Mumford stack is an essentially trivial gerbe on its rigidification. In Section 6.3, we will prove the main result in Theorem 6.25.

6.1. Gerbes and root constructions. First, we recall some general notion on banded gerbes (gerbes lieés). We refer to [18], chapter IV.2, for a complete treatment and to [16], Section 3, for a shorter reference. Let X be a smooth Deligne-Mumford stack. Let G be an abelian sheaf of groups3) and G X a gerbe. For every e´tale chart U of X and every ob- G ! ject x A U let ax : G U AutU x be an isomorphism of sheaves of groups such that the natural compatibilities j coming! from the fibered structure of the gerbe are satisfied. The collection of these isomorphisms is called a G-banding.AG-banded gerbe is the data of a gerbe and a G-banding. Two G-banded gerbes are said to be G-equivalent if they are isomorphic as stacks and the isomorphism makes the two bandings compatible in the natural way. Gir- 2 X aud proved in [18] (Chapter IV, 3.4) that the group Het´ ; G classifies equivalence classes of G-banded gerbes.

Remark 6.1. We anticipate some observations about the banding which will be useful in the following:

(1) The b-th root of a line bundle on X is a gerbe which is banded in a natural way by the constant sheaf mb; the banding is the canonical isomorphism between the group of auto- morphisms of any object and mb.

(2) Given G X a G-banded gerbe, every rigidiﬁcation of G by a subgroup H of G inherits a G=H -banding! from the G-banding of G. Here we introduce the concept of an essentially trivial gerbe which will play an important role in this section. The Kummer sequence

i 5b 1 m Gm Gm 1 ! b ! ! !

3) The non-abelian case has a richer structure but for the sake of simplicity we just skip all these additional features and refer the interested reader to [18].

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 221 induces the long exact sequence

1 q 2 i 2 6:2 H X; Gm H X; m H X; Gm : ! et´ ! et´ b! ! 2 X Deﬁnition 6.3. A mb-banded gerbe in Het´ ; mb is essentially trivial if its image by i 2 is the trivial gerbe in H X; Gm . et´

Remark 6.4. (1) It follows from Section 1.3.a that a mb-banded gerbe is essentially trivial if and only if it is a b-th root of an invertible sheaf on X.

b nb X b X (2) As the mb-banded gerbe L n M = is isomorphic to L= , we deduce a bijection between essentially trivialpm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi-banded gerbes and Pic X =b PicpffiffiffiffiffiffiffiffiffiffiX . b 2 X Lemma 6.5. There is a natural bijection between essentially trivial gerbes in Het´ ; mb and elements in Ext1 Z=bZ; Pic X . Proof. By Remark 6.4(2), it is enough to show that Ext1 Z=bZ; Pic X is isomorphic to Pic X =b Pic X . This follows from the exact sequence 5b Hom Z; Pic X Hom Z; Pic X Ext1 Z=b; Pic X 0: r ! ! ! l

Let G be a ﬁnite abelian group. Fix a decomposition G mbj . We deduce an iso- j 1 morphism Q

l 2 X 2 X 6:6 Het´ ; G Het´ ; mbj ; !j 1 L a a ; ...; al : 7! 1 Deﬁnition 6.7. Let G be a ﬁnite abelian group. A G-banded gerbe associated to l A 2 X a H ; G is essentially trivial if there is a decomposition of G mbj such that for any l j 1 j A 1; ...; , the mb -banded gerbe aj is essentially trivial. Q f g j Remark 6.8. Being essentially trivial does not depend on the choice of a decomposition of G. l

Proposition 6.9. Let G be a ﬁnite abelian group. Fix a decomposition of G mbj . j 1 There are bijections between Q

l 2 X Essentially trivial gerbes in Het´ ; mbj j 1 L 1:1 Fibered products over X of bj-th roots of invertible sheaves $ f g l l 1:1 1:1 1 Pic X =bj Pic X Ext Z=bjZ; Pic X : $ j 1 $j 1 Q Q

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222 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Remark 6.10. To be more concrete, let us explicitly describe the last bijection. For the sake of simplicity, we consider the case j 1. To the class L0 in Pic X =b Pic X ,we associate the extension

0 Pic X Pic X Pic X =b Pic X Z=bZ Z=bZ 0 ! ! ! ! where the ﬁber product is given by the standard projection Pic X Pic X =b Pic X and ! the morphism Z=bZ Pic X that sends the class of 1 to the class L0 . The ﬁrst morphism in the extension sends! the invertible sheaf L to Lnb; 0 . Let 0 Pic X A Z=b 0 be an extension. We consider the projective resolu- !b ! ! ! tion 0 Z Z Z=b 0. There exists f and ff~ such that the following diagram is a morphism! of! short! exact! sequences:

0 Z Z Z=b 0 ! ! ! ! ff~ f ? ? ? ? 0 Pic?X A? Z=b 0: ! y ! y ! ! The class ff~ 1 in Pic X =b Pic X is the element that corresponds to the above extension. Notice that di¤erent liftings f , ff~ lead to di¤erent elements in Pic X with the same class in Pic X =b Pic X . The two maps deﬁned above are inverse to each other.

Proof of Proposition 6.9. Most of the proposition is a direct consequence of Remark 6.4 and Lemma 6.5. The only non-trivial fact to prove is that an essentially trivial gerbe l 2 a a ... al A H X m defined by 1; ; et´ ; bj is given by a fiber product of the gerbes defined j 1 L by the ajs. Without loss of generality, we can assume that a a ; a ; the general case is 1 2 proved by induction. The gerbe defined by a1 (resp. a2) is isomorphic to the rigidification

G m G m

( b2 (resp. ( b1 ). Hence we have the following 2-commutative diagram:

G m

( b2

! GX !

! G m ( b1 :

G G m G G m m Remark 6.1(2) implies that ( b1 (resp. ( b2 )isa b2 -banded gerbe (resp. m ! ! b1 -banded). By the universal property of the ﬁber product we are given a morphism G G m X G m X ( b1 ( b1 . Two gerbes banded by the same group over the same base are! either isomorphic as stacks or they have no morphisms at all; this completes the proof. r

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 223

6.2. Gerbes on toric orbifolds.

Theorem 6.11. Let X be a toric orbifold with torus T. Denote by i : T , X the immersion of the torus. Then the morphism !

2 2 i : H X; Gm H T; Gm et´ ! et´ is injective.

Notice that in the following proof we will use that a toric orbifold is a global quotient ZS=GX where GX : HomZ Pic X ; C . This will be proved in Theorem 7.7 and does not depend on the results of this subsection.

We ﬁrst proof some preliminary results.

Lemma 6.12 (Artin). Let S be a smooth quasi-projective variety, S2 L S a closed i G i G subscheme of codimension f 2. Then the natural map Het´ S; m Het´ S S2; m is an isomorphism for all i. ! n

Proof. The statement is obvious if we replace sheaf cohomology with Cˇ ech cohomology. To prove the lemma, we just apply [6], Corollary 4.2, p. 295 (see also [28], Theo- rem 2.17, p. 104). r

Lemma 6.13 (Olsson). Let X be an Artin stack and X0 an atlas. Denote by Xp X0 X X X0. Let F be an abelian sheaf of groups on X and Fp its restriction to pq q p q Xp. There is a spectral sequence with E X : H Xp; Fp that abuts to H X; F . 1 et´ et´ Proof. This lemma follows immediately from [30], Corollary 2.7, p. 4 and Theorem 4.7, p. 13. r

Proof of Theorem 6.11. Let X be a toric orbifold with coarse moduli space a simplicial toric variety X. Denote by S H NQ the fan of X. Without lost of generality, we can assume that the rays of S generate NQ.

By Theorem 5.2 in the case of orbifolds and Lemma 7.1, we have that X ZS=GX where GX : HomZ Pic X ; C . Denote by n the number of rays of the fan S. Put n Z : z A ZS H C Ei A 1; ...; n ; zj 0 2 j f g jQ3i the union of T-orbits in ZS of codimension f 2. The closed subscheme Z2 of ZS is of codimension 2. Hence the quotient stack ZS Z2 =GX is a closed substack of codimension 2 of X. For any i A 1; ...; n , put n f g n Ui : z A ZS H C Ej A 1; ...; n i ; zj 3 0 : f j f gnf g g

1 n 1 We have that Ui is isomorphic to A C and that the natural morphism

Ui ZS Z2 i A 1;...; n ! n f g

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224 Fantechi, Mann and Nironi, Deligne-Mumford stacks is e´tale and surjective. We deduce that Ui=GX ZS Z =GX is e´tale and surjec- i A 1;...; n ! n f g tive. Put X0 : Ui. The natural morphism X0 ZS Z2 =GX is an e´tale atlas. i A 1;...; n ! n f g

Denote Xp X0 X X X0. From Lemma 6.13 we have a spectral sequence pq q p q E X : H Xp; Gm abutting to H ZS Z2 =GX ; Gm . Using this spectral se- 1 et´ jXp et´ n quence and Lemma 6.12 we obtain that the natural morphism

i i H X; Gm H ZS Z =GX ; Gm et´ et´ n 2 is an isomorphism for i 0; 1; 2 . Finally, the theorem follows from Lemmas 6.14 and 6.15. r

Lemma 6.14. We have the following morphism of short exact sequences:

20 2 02 0 E X H X; Gm E X 0 ! 4 ! et´ ! 2 !

a j b ? ? ? ? ? ? 20 2 02 0 E ?T H T?; Gm E ?T 0: ! 4 y ! et´ y ! 2 y ! 20 X 20 T 02 X 02 T Lemma 6.15. The vertical maps a : E4 E4 and b : E2 E2 are injective. ! !

pq Proof of Lemma 6.14. To prove the lemma, we are just interested in Ey X for p q 2. We start by proving that we have

20 2 02 6:16 0 E X H X; Gm E X 0: ! y ! et´ ! y !

Hilberts Theorem 90 (cf. [28], Proposition 4.9) implies that

1 1 H Xp; Gm H Xp; O Pic Xp : et´ Zariski Xp

Using the notation of the proof of Theorem 6.11, for any ray i A 1; ...; n , we have that 1 n 1 f g Ui=GX is isomorphic to A =m C where ai is the multiplicity along the divisor Di ai (cf. Remark 3.7). Hence, we have that Xp Ui ip where 0 i ;...; ip A 1;...; n 0 f g

U mp 1 if i i ; i0 a0 0 p Ui0 ip T otherwise:

p1 X p1 X 1 G Hence, for any p we have that E1 Ey Het´ Xp; m 0. We deduce the exact sequence (6.16).

We now show that E 20 X E 20 X and E 02 X E 02 X . y 4 y 2

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 225

Figure 1. Terms E pq X and E pq X 1 2

Figure 2. Terms E pq X and E pq X 3 4

In Figures 1 and 2, the circled terms mean that they will stay constant that is they are pq 20 20 02 02 equal to Ey X . We deduce that E X E X and E X E X . y 2 y 4 The same argument for T proves the lemma. r

20 X 20 Proof of Lemma 6.15. First, we show that the morphism a : E4 E4 T is injective. From Figures 1 and 2, we have that !

6:17 E 20 X ker d : E 20 X E 03 X ; 4 3 2 ! 2 6:18 E 20 T ker d : E 20 T E 03 T : 4 3 2 ! 2 Moreover, we have that

20 2 2 6:19 E X ker d : H X ; Gm H X ; Gm ; 2 1 et´ 0 ! et´ 1 20 2 2 6:20 E T ker d : H T ; Gm H T ; Gm : 2 1 et´ 0 ! et´ 1 1 n 1 n Recall that Ui F A C and T0 C . By Grothendiecks Expose´s on the Brauer group [20], §6, p. 133, we have the following long exact sequence:

2 G 2 G 2 G 6:21 HX T X0; m Het´ X0; m Het´ T0; m : ! 0 0 ! ! !

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226 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Moreover, we have that:

pq p q G The spectral sequence F2 : H X0 T0 ; H X T X0; m converges to n 0n 0 H X T X0; Gm . 0n 0 0 2 1 H X ; Gm H X ; Gm 0 and H X ; Gm Z. X T 0 X0 T0 0 X0 T0 0 0n 0 n n 20 02 n 1 This implies that F F 0. As X T C , we have that 2 2 0n 0 F 11 H 1 X T ; Z 0: 2 0n 0

pq 2 The spectral sequence F implies H X0; Gm 0. Hence, sequence (6.21) and 2 X0 T0 equalities (6.17), (6.18), (6.19) and (6.20), implyn that a is injective.

02 X 02 02 X Let us prove that b : E2 E2 T is injective. Recall that E2 ker d2=Im d1 and E 02 T ker dd~ =Im dd~ . We ! have the following commutative diagram: 2 2 1

0 d1 0 d2 0 H X1; Gm H X2; Gm H X3; Gm ! ! L L L

~ ~ 0 dd1 0 dd2 0 H T1; Gm H T2; Gm H T3; Gm : ! ! As Tii 0 , Uii 0 is open and dense, the vertical maps are injective. Notice that these maps ! 0 are isomorphisms except on Uii and Uiii. Let yy~ A H Tii; Gm such that there exists 0 ~ x A H Uiii; Gm that lifts dd yy~ , i.e., we have the following diagram: 1 x !

~ yy~ dd1 yy~ : ! ~ 0 0 0 The morphism dd2 : H Tii; Gm H Tiii; Gm is defined, for any yy~ A H Tii; Gm and jTii ! any t; g; h A Tiii Ti m m ,by ai ai ~ dd2 yy~ t; g; h yy~ ht; g yy~ t; h =yy~ t; gh : jTii A The divisor Ui Ti is a principal divisor associate to the rational function j. For any g mai , n N the function yy~ is rational on Uii g Ui g . Hence there exists a unique n g in such n g j f g that yy~j is a regular function on Ui g .Asyy~ ht; g yy~ t; h =yy~ t; gh is a regular function, we deduce that jn g n h n gh 1.f Hence,g the function n : m Z is a group homo- ai ! morphism, therefore n g 1 for every g. We deduce that yy~ is a regular function on Ui which implies that the morphism b : E 02 X E 02 T is injective. r 2 ! 2 6.3. Characterization of a toric Deligne-Mumford stack as a gerbe over its rigidification. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T isomorphic to T BG and coarse moduli space X. Denote by Xrig the rigidification of X (cf. Sec- tion 1.4) which is by definition an orbifold with coarse moduli space X. The universal

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 227 property of the rigidiﬁcation and of the canonical stack (see Proposition 1.5 and Corollary 4.10) imply that we have the following strictly commutative diagram:

X r Xrig ! 6:22 f rig ? f ? X?can y : Section 1.4 and Lemma 3.8 imply that we can deﬁne X( G.

Lemma 6.23. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T isomorphic to T BG. (1) The orbifold Xrig is canonically isomorphic to X( G.

(2) There is a unique structure of toric orbifold on Xrig with torus T such that the morphism r : X Xrig is a morphism of toric Deligne-Mumford stacks induced by T T. ! ! Remark 6.24. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T isomorphic to T G and coarse moduli space X. (1) Proposition 5.1 implies that the morphism f rig : Xrig Xcan is a morphism of toric Deligne-Mumford stacks. Hence we deduce that the commutative! diagram (6.22) is a commutative diagram of morphisms of toric Deligne-Mumford stacks.

(2) Let H be a subgroup of G. The stack X( H is a toric Deligne-Mumford stack with Deligne-Mumford torus isomorphic to T( H F T B G=H . Moreover, the natural morphisms X X( H and X( H X( G are morphisms of toric Deligne-Mumford stacks. ! !

(3) Note that we did not use the non-canonical isomorphism T 1 T BG but only the short exact sequence of Picard stacks 1 BG T T 1. ! ! ! ! Proof of Lemma 6.23. (1) As T( G is isomorphic to the scheme T which is open and dense in X( G, the stack X( G is an orbifold which is canonically isomorphic to Xrig.

(2) The morphisms i : T , X and a : T X X induce morphisms on the rigid- iﬁcations irig : T( G F T Xrig!and arig : T Xrig ! Xrig, by the universal property of the rigidiﬁcation (see Proposition! 1.5). It is immediate ! to verify that arig is an action, ex- 1 tending the action of T on itself. As r T is isomorphic to T, we deduce that this is the only toric structure on Xrig which is compatible with the morphism r. r

Since the morphism r : X Xrig is e´tale, the divisor multiplicities of X and Xrig are the same. !

Theorem 6.25. (1) Let Y be a toric orbifold with Deligne-Mumford torus T. Let X Y be an essentially trivial G-gerbe. Then X has a unique structure of toric Deligne- Mumford! stack with Deligne-Mumford torus isomorphic to T BG such that the morphism X Y is a morphism of toric Deligne-Mumford stacks. !

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228 Fantechi, Mann and Nironi, Deligne-Mumford stacks

(2) Conversely, let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T F T BG. Then X Xrig is an essentially trivial G-gerbe. ! Proof. (1) The inverse image of T in X, denoted by T, is open dense. The restriction of the essentially trivial G-banded gerbe X Y to T is the essentially trivial G-banded gerbe T T. Remark 6.4(1) implies that the gerbe! T T is trivial. The action of T on Y induces! by pullback an action of T on X. This is the! only structure of toric Deligne- Mumford stack on X compatible with the morphism X Y. ! 2 Xrig X Xrig (2) Denote by a A Het´ ; G the G-banded gerbe . By Proposition 2.6, the restriction of a on the Deligne-Mumford torus T is the! trivial G-banded gerbe in l H 2 T G G m a et´ ; . Fix a cyclic decomposition of bj . By the isomorphism (6.6), the class l j 1 rig Q a ... al A H 2 X m j A ... l is sent to 1; ; et´ ; bj . We have that for any 1; ; , the class of j 1 f g L a H 2 T m i j restricts to the trivial class in et´ ; bj . Theorem 6.11 states the injectivity of in the following diagram:

bj p =X rig 1 rig 2 rig 2 rig H X ; Gm H X ; m H X ; Gm et´ ffiffiffiffiffiffiffiffiffi! et´ bj ! et´ L

i ? ? ? ? ? 2 ? 2 G y1 H´ Ty; mbj H´ T; m : ! et ! et A simple diagram chasing ﬁnishes the proof. r

Corollary 6.26. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus T isomorphic to T BG. l Xrig X (1) Given G mbj . There exists Lj in Pic such that is isomorphic as G- j 1 Q banded gerbe over Xrig to

b1 rig bl rig L1=X X rig X rig Ll=X : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l rig rig Moreover, the classes L1 ; ...; Ll in Pic X =bj Pic X are unique. j 1 Q (2) The reduced closed substack X T is a simple normal crossing divisor. n The first part of the corollary is very similar to [31], Proposition 2.5.

Remark 6.27. Let X be a toric Deligne-Mumford stack with Deligne-Mumford l T B torus isomorphic to T G and G mbj . Diagram (1.3) and the corollary above j 1 Q imply that we have the following morphism of short exact sequences:

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 229

l l b l 0 Z Z Z=bjZ 0 ! ! ! j 1 ! L 6:28 ? ? ? ? l ? rig r ? 0 Pic yX PicyX Z=bjZ 0 ! ! ! j 1 ! L 1=bj where the vertical morphisms sends ej Lj and ej L . 7! 7! j Proof of Corollary 6.26. Theorem 6.25(2) implies that X Xrig is an essentially trivial G-banded gerbe. The ﬁrst statement follows from Proposition! 6.9.

By Corollary 5.4, we have that the reduced closed substack Xrig Trig is a simple normal crossing divisor. As the morphism X Xrig is e´tale, we deducen the second statement of the corollary. r !

7. Toric Deligne-Mumford stacks versus stacky fans

In this section, we will show that the toric Deligne-Mumford stacks that we have deﬁned correspond exactly with those of [10].

In the ﬁrst subsection, we show that our toric Deligne-Mumford stacks with a spanning condition are global quotients. The second subsection makes the correspondence with the article of [10].

7.1. Toric Deligne-Mumford stacks as global quotients. Let Z be a subvariety in Cn of codimension equal or higher than two. Let G be an abelian group scheme over C that acts on Z such that Z=G is a Deligne-Mumford stack. According to Remark 1.1, a line bundle on Z=G is given by a character w of G. Hence the data of an invertible sheaf L with 1 a global section s on Z=G give a morphism of groupoids between Z=G and A =C . 1 1 Explicitly, this morphism is given by s; w : Z G A C and s : Z A . ! ! In the following lemma, we use a slightly more general notion of a root of Cartier divisors that is a root of invertible sheaves with global sections. All the properties of Section 1.3.b are still true (see [12] or [2]).

Lemma 7.1. Let Z be a scheme. Let G be an abelian group scheme over C that acts on Z such that Z=G is a Deligne-Mumford stack. Let L; s : L ; s ; ...; Lk; sk 1 1 be k invertible sheaves with global sections over the quotient stack Z=G . Denote by L w : w1; ...; wk the representations associated to the invertible sheaves . Let k d : d ; ...; dk be in N . 1 >0 (1) We have that d L; s = Z=G is isomorphic to ZZ~=GG~ where ZZand~ GGare~ defined by the following cartesianp diagramsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi:

ZZ~ Ak GG~ Gk ! ! m rr5d j 5d ? ? ? ? ? s ? ? w ? Z? A?k; G? G?k : y ! y y ! ym AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR

230 Fantechi, Mann and Nironi, Deligne-Mumford stacks

The action of GGon~ ZZis~ given by

g; l ; ...; lk z; x ; ...; xk gz; l x ; ...; lkxk 1 1 1 1 ~ for any g; l ; ...; lk A GGand z; x ; ...; xk A ZZ~. 1 1 d (2) We have that L= Z=G is isomorphic to Z=GG~ where GGis~ defined above. The action of GGon~ Z is givenp viaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij. Proof. It is a straightforward computation on fibered products of groupoids. r

Remark 7.2. (1) We have that ker j is isomorphic to mdi . Notice that the action of i 1 Q GG~ on Z in the second part of the proposition above implies that the kernel of j acts trivially k ~ on Z. Hence, Z=GG is a mdi -banded gerbe over Z=G . i 1 Q k ~ A 1 (2) In both cases we have that GG Ext G; mdi . i 1 Q k 1 Lemma 7.3. Let A be an abelian group of ﬁnite type. Let E in Ext Z=diZ; A .If i 1 we have a morphism of short exact sequences L k k d1;...; dk k 0 Z Z Z=diZ 0 ! ! ! i 1 ! L ? ? ?o ? ? k ? ? ? ? 0 Ay Ey Zy=diZ 0 ! ! ! i 1 ! L then the left square is cocartesian. Remark 7.4. Diagrams (5.6) and (6.28) imply that we have the following cocartesian diagrams:

l b l n a n Z Z Z Z ! ! 7:5 ? ? ? ? ? ? ? ? ? b? ?X a?D X Pic X Pic L=X ; Picy Pic y = : y ! y ! pffiffiffiffiffiffiffiffiffiffi k k pffiffiffiffiffiffiffiffiffiffiffik Proof of Lemma 7.3. Denote by P the push-out ofZ Z and Z A. Using the universal property of co-cartesian diagrams we deduce a morphism! f from!P to E and the following morphisms of extensions: k k d1;...; dk k 0 Z Z Z=diZ 0 ! ! ! i 1 ! L ? ? ?a ? ? ? ? q ? ? 0 A? P? coker? q 0 ! y ! y ! y ! f ? ?b ? ? ? k ? ? ? 0 A Ey Zy=diZ 0: ! ! ! i 1 ! L

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 231

Notice that the composition b a is the isomorphism in Lemma 7.3. By simple diagram chasing, we deduce that f is an isomorphism. r

Remark 7.6. Let X be a toric Deligne-Mumford stack with coarse moduli space X. Proposition 3.6 implies that X is a simplicial toric variety. Denote by S a fan of X. Assume that the rays of S generate NQ. As explained in Section 1.6, we have that X is the geometric 1 quotient ZS=GA where GA : Hom A X ; C . Put GX : Hom Pic X ; C . Notice that n rig GX rig acts on ZS via the dual (in the sense of Section 1.5) of the morphism Z Pic X . rig ! The group GX acts on ZS via the dual of the morphism Pic X Pic X . Consider the n ! quotient stack ZS=GX . The quotient stack C =GX is a Deligne-Mumford torus which n is open and dense in ZS=GX . As the natural action of C on ZS extends the action of n n C on itself, we deduce a stack morphism a : C =GX ZS=GX ZS=GX that ex- n ! tends the action of C =GX on itself. Proposition 3.3 implies that the stack morphism a induces a natural action of the Deligne-Mumford torus on ZS=GX that is ZS=GX is a toric Deligne-Mumford stack.

Theorem 7.7. Let X be a toric Deligne-Mumford stack with coarse moduli space X. Denote by S the fan associated to X. Assume that the rays of S generate N n Q. Then X is naturally isomorphic, as a toric stack, to ZS=GX where GX : Hom Pic X ; C . Remark 7.8. Removing the spanning condition of the rays gives the following result. Let X be a toric Deligne-Mumford stack with torus T (isomorphic to T BG) and with coarse moduli space the simplicial toric variety X. Denote by S H NQ the fan of X. From the footnote 2 of Section 1.6, we deduce that the toric variety X is isomorphic to XX~ TT~ where XX~ is a simplicial toric variety whose the rays of its fan SS~ span NN~ Q. Notice that the dimension of TT~ is rk NQ rk NN~ Q . The previous theorem implies that X is isomorphic, as toric stacks, to ZSS~=GXX~ TT~ BG . Proof of Theorem 7.7. If X is Xcan, the theorem follows from Remark 4.12(2). If X is Xrig, the theorem follows from the right cocartesian square of diagram (7.5) and Lemma 7.1(1). For a general X, it follows from the left cocartesian square of diagram (7.5) and Lemma 7.1(2). r

7.2. Toric Deligne-Mumford stacks and stacky fans. First we recall the deﬁnition of a stacky fan from [10].

Deﬁnition 7.9. A stacky fan is a triple S : N; S; b where N is a ﬁnitely generated abelian group, S is a rational simplicial fan in NQ : N nZ Q with n rays, denoted by r ; ...; r , and a morphism of groups b : Zn N such that: 1 n !

(1) The rays span NQ.

(2) For any i A 1; ...; n , the element b ei in NQ is on the ray ri where e1; ...; en is f n g the canonical basis of Z and the natural map N NQ sends m m. ! 7! Remark 7.10. Let S : N; S; b be a stacky fan.

(1) As the rays span NQ, we have that b has ﬁnite cokernel.

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232 Fantechi, Mann and Nironi, Deligne-Mumford stacks

(2) For any i A 1; ...; n , denote by vi the unique generator of ri X N=Ntor f g rig where Ntor is the torsion part part of N. Denote by b the composition of b followed by the quotient morphism N N=Ntor. There exists a unique ai A N>0 such that rig rig ! rig b ei aivi. Denote S : N=Ntor; S; b . There exists a unique group homomorphism b can : Zn N=N such that we have the following commutative diagram:

! tor

Zn N ! b rig

7:11 diag a1;...; an ? ? ? ? can ?n b ? Z? N=?Ntor: y ! y Denote Scan : N=N ; S; b can . tor In [10], Remark 4.5, the authors deﬁne the notion of morphism of stacky fans. The commutative diagram (7.11) provides us the morphisms of stacky fans S Srig Scan. ! ! (3) To the fan S, we can associate canonically the stacky fan Scan.

Construction 7.12 (construction of the Deligne-Mumford stack associated to the stacky fan S). Now we explain how to associate a Deligne-Mumford stack X S to a stacky fan S following [10], Sections 2 and 3. Denote by d the rank of N. Choose a projective resolution of N with two terms that is

l Q d l 0 Z Z N 0: ! ! ! !

n d l n Choose a map B : Z Z lifting the map b : Z N. Consider the morphism n l d l ! ! 4 n BQ : Z Z . Denote DG b : coker BQ . Denote by b : Z DG b the

group morphism! that makes the following diagram commute: !

Zn K Zn l

! 4 b ? ? DG b : ?coker BQ : y

Let ZS be the quasi-a‰ne variety associated to the fan S (see Section 1.6). Deﬁne the action of GS : HomZ DG b ; C on ZS as follows. Applying the functor HomZ ; C 4 n n to the morphism b : Z DGb , we get a group morphism GS C . Via the nat- n n ! ! ural action of C on C , we deﬁne an action of GS on ZS. Finally, the stack associated to the stacky fan S : N; S; b is the quotient stack X S : ZS=GS .

Notation. We will later see that the group GS is isomorphic to

GX : Hom Pic X ; C : By [10], Proposition 3.2, we have that ZS=GS is a smooth Deligne-Mumford stack.

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 233

Remark 7.13. In [24], Iwanari deﬁned a smooth toric Artin stack over any scheme associated to a stacky fan Srig.

Remark 7.14. As it was observed in [10], Section 4, the condition that the rays span d NQ in Deﬁnition 7.9 is not natural. Indeed a Deligne-Mumford torus C BG where G is a ﬁnite abelian group can not be produced as a stack X S for S a stacky fan with the condition that the rays span NQ. Nevertheless, it is not really true to say that toric Deligne- Mumford stacks are a ‘‘generalization of the stacks X S . Indeed, as for toric variety, we will see that a toric Deligne-Mumford stack is a product of a X S by a Deligne-Mumford torus.

Lemma 7.15. Let S : N; S; b be a stacky fan. (1) The stack X S is a toric Deligne-Mumford stack. (2) The stack X S is a toric orbifold if and only if the ﬁnitely generated abelian group N is free.

(3) The stack X S is canonical if and only if S Scan. n Proof. (1) The group morphism GS C defined in Construction 7.12 defines n ! the quotient stack C =GS which is by definition a Deligne-Mumford torus. As the n open dense immersion C , ZS is GS-equivariant, we have that the stack morphism n ! C =GS ZS=GS is an open dense immersion. Using the same arguments of Remark ! n 7.6, we have that the action of the Deligne-Mumford torus C =GS on itself extends to an action on ZS=GS . That is X S is a toric Deligne-Mumford stack. n (2) The stack X S is a toric orbifold if and only if GS C is injective, if and only if b4 is surjective, if and only if N is free. !

(3) Assume that S Scan. As the coarse moduli space X of X S is the geometrical 1 can quotient ZS=GA1 X where GA1 X : Hom A X ; C , we have that X ZS=GA1 X . S Scan Construction 7.12 implies that GS is GA1 X . Conversely, if 3 then either N has torsion (i.e., X S is a gerbe) or there exists a divisor D associated to a ray such that any geometric point of D has a non-trivial stabilizer. r

Remark 7.16. Let X S be a canonical stack (i.e., S Scan). The proof of the third statement of Lemma 7.15 implies that DG b can Pic X S . Theorem 7.17. Let X be a toric orbifold with coarse moduli space X. Denote by S a fan of X in NQ : N nZ Q. Assume that the rays of S span NQ. Then there is a unique b : Zn N such that the stack associated to the stacky fan N; S; b is isomorphic as toric orbifold! to X.

k Remark 7.18. An arbitrary toric orbifold is isomorphic to a product X S C .

Proof of Theorem 7.17. Denote by a : a1; ...; an the divisor multiplicities of X. n We deﬁne the morphism of groups b : Z N by sending ei aivi where vi is the generator of the semi-group r X N. Denote by S!the stacky fan N;7!S; b . i

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234 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Theorem 7.7 states that X is isomorphic to ZS=GX . In order to prove that the two stacks are isomorphic, we will show that GX is isomorphic to GS such that the two actions on ZS are compatible. From diagram (7.11), we deduce a morphism of exact sequences:

n n n 0 Z Z Z=aiZ 0 ! ! ! i 1 ! L ? ? ? ? n X? can ? Z Z 0 Pic y DGy b =ai 0: ! ! ! i 1 ! L The right cocartesian square of diagram (7.5) implies that GS is isomorphic to GX such that the actions of GS and GX on ZS are compatible.

The uniqueness of b follows from the geometrical interpretation of the divisor multiplicities. r

Remark 7.19. (1) The proof shows also that Pic X is isomorphic to DG b .

(2) Marking a point aivi on the ray ri X N corresponds geometrically to putting a D generic stabilizer mai on the divisor i associated to the ray ri.

Proposition 7.20. Let S : N; S; b be a stacky fan. There is a unique a in 1 rig Ext N ; Pic X S such that the essentially trivial Hom N ; C -banded gerbe over tor tor X Srig associated to ais isomorphic as banded gerbe to X S . l d Proof. Fix a decomposition N Z l Z=bjZ. It follows from Construction 7.12 j 1 that we have the following diagram: L

000

x x x ? ? l ? ? rig ? Z? Z 0 DG?b DG? b ?=bj 0 ! ! ! j 1 ! L b rig 4 x x x ?n n?l ?l 7:21 0 Z? Z ? Z? 0 ! ? ! ? ! ? !

BQ b1;...;bl x x x ? ? ? d d l l 0 Z? Z ? Z? 0 ! ? ! ? ! ? ! x x x ? ? ? ?000? ?: ? ? ? From Remark 7.19, we have that Pic X Srig is isomorphic to DG b rig . The ﬁrst line 1 rig of diagram (7.21) is an element a AExt Ntor ; Pic X S . By Proposition 6.9, the l rig rig element a induces an element L1 ; ...; Ll A Pic X =bj Pic X . The last row of j 1 Q

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 235

l the diagram above is a projective resolution of Z=bj. Hence, we deduce that there exists j 1 a morphism of short exact sequences L

l l b l 0 Z Z Z=bjZ 0 ! ! ! j 1 ! L 7:22 ff~ f ? ? ? ? l ?Xrig r ? 0 Pic y DGy b Z=bjZ 0: ! ! ! j 1 ! L rig The morphism ff~ is the same as the choice of L ; ...; Ll in Pic X in the classes 1 L ; ...; Ll . By the left cocartesian square of diagram (7.5), we deduce that GS is isomor- 1 phic to GX. We conclude that X is isomorphic to X S . The uniqueness of a follows from Proposition 6.9. r

Remark 7.23. Denote by X1 and by X2 respectively the stacks associated to S S X X stacky fans ; N; b1 and ; N; b2 . The stacks 1 and 2 are isomorphic, as toric Deligne-Mumford stack, if and only if the extensions deﬁned in diagram (7.21) in Ext1 N ; Pic X Srig are isomorphic. tor Theorem 7.24. Let X be a toric Deligne-Mumford stack with coarse moduli space X. Denote by S a fan of X in NQ. Assume that the rays of S span NQ. There exist N and b : Zn N such that the stack associated to the stacky fan N; S; b is isomorphic as toric Deligne-Mumford! stacks to X.

Remark 7.25. Let S be a stacky fan. Corollary 6.26 and the theorem above imply that X S is isomorphic to a product of root stacks over its rigidiﬁcation. This result was discovered independently by Perroni (cf. [31], Proposition 3.2) and by Jiang and Tseng (cf. [25], Remark 2.10).

Proof of Theorem 7.24. If X is a toric orbifold then the statement was already proved in Theorem 7.17.

Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus isomorphic to T BG. By Theorem 7.7, we have that X is isomorphic to ZS=GX . By Theorem 7.17, there exists a unique stacky fan Srig S; Zd ; b rig where d : dim X such that Xrig is isomorphic to X Srig . l l l d ... l A N : Z l Z Z There exist b1; ; b >0 such that G mbj . Put N =bj . j 1 j 1 Q rig L Corollary 6.26 gives us l invertible sheaves L1; ...; Ll on X . For any j, choose n Z O Drig cij Drig c1j; ...; cnj A such that Lj i where i is the Cartier divisor associated i 1 to the ray ri. Put N

l n d b : Z Z l Z=bjZ; ! j 1 L rig ei b ei ; ci ; ...; cil ; 7! 1

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236 Fantechi, Mann and Nironi, Deligne-Mumford stacks where cij is the class of cij modulo bj. It is straightforward to check that X S is isomorphic to X. r

Remark 7.26. Let X be a toric Deligne-Mumford stack with Deligne-Mumford torus isomorphic to T BG. The non-uniqueness of N and b comes from three di¤erent kinds: l

(1) the decomposition of G in product of cyclic groups, i.e., G mbj , j 1 Q rig rig (2) the choice of the lift for the class Lj A Pic X =bj Pic X for j 1; ...; l, and n O Drig cij (3) the choice of the decomposition Lj i (see Example 7.29 for such an i 1 example). N

7.3. Examples.

Example 7.27 (weighted projective spaces). Let w0; ...; wn be in N>0. Denote n 1 by P w the quotient stack C 0 =C where the action of C is deﬁned by w0 wn nf g n 1 l x0; ...; xn l x0; ...; l xn for any l A C and any x0; ...; xn A C 0 . The stack P w is a complete toric Deligne-Mumford stack with Deligne-Mumford nf g torus n 1 n C =C F C Bm where d : gcd w ; ...; wn (cf. Example 2.3). d 0 We have that:

(1) The stack P w is canonical if and only if for any i A 0; ...; n , we have that f g gcd w ; ...; ww^i; ...; wn 1 (e.g., the weights are well-formed). 0

(2) The stack P w is an orbifold if and only if gcd w ; ...; wn 1. 0 (3) The Picard group of P w is cyclic. More precisely, we have Z if dim P w f 1; Pic P w Z=w Z if P w P w : 0 0 Proposition 7.28. Let X be a complete toric Deligne-Mumford stack of dimension n such that its Picard group is cyclic. Then there exists unique up to order w0; ...; wn in n 1 N such that X is isomorphic to P w ; ...; wn . >0 0 Proof. Denote by X the coarse moduli space of X. Denote by S a fan of X.If X the Picard group is isomorphic to Z=dZ then Theorem 7.7 implies that ZS=md with n ZS H C . Hence, the fan S has n rays. In this case, X is complete if and only if n 0. We deduce that X Bm F P d . d n 1 If the Picard group is Z, Theorem 7.7 implies that X ZS=C with ZS H C . n 1 As X is complete, the fan S is complete. We deduce that ZS C 0 . The Deligne- n 1 nf g Mumford torus is isomorphic to C =C . The action of C is given by the morphism n 1 w0 wn C C that sends l l ; ...; l with wi A Z 0 . Notice that if the wis do not ! 7! nf g

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 237 have the same sign then X is not separated. If the wis are all negative then replacing l by 1 l induces an isomorphism with a weighted projective space. r

Example 7.29. In this example, we give two isomorphic stacky fans for P 6; 4 which was considered in [10], Example 3.5. As we have seen in Section 7.2, N and S are ﬁxed whereas b is not unique. Let N be Z Z=2. Let S be the fan in NQ Q where the cones are 0, Qf0, Qe0. Put

7:30 b : Z2 Z Z=2; b : Z2 Z Z=2; 1 ! 2 ! e 2; 1 ; e 2; 1 ; 1 7! 1 7! e 3; 0 ; e 3; 1 : 2 7! 2 7! One can check that the stack associated to N; S; b and N; S; b is P 6; 4 . 1 2 Let us explicit the bottom up construction in this case. Its coarse moduli space is P1. The rigidification of P 6; 4 is P 3; 2 . Denote by x1, x2 the homogeneous coordinates of 2; 3 1 1 P . We have that P 3; 2 D1; D2 =P where Di is the Cartier divisor OP1 1 ; xi . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We have that OP 3; 2 D1 OP 3; 2 3 , OP 3; 2 D2 OP 3; 2 2 and pOP1 1 OP 3; 2 6 P P 1 P P where p : 3; 2 is the structure morphism. The stack r : 6; 4 3; 2 is a m2- ! 2 ! banded gerbe isomorphic to OP 3; 2 1 =P 3; 2 . In Pic P 3; 2 =2 Pic P 3; 2 , the class of OP 3; 2 1 is also the class ofpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiOP 3; 2 D1 or the class of OP 3; 2 D1 nOP 3; 2 D2 . These di¤erent choices lead to the two isomorphic stacky fans in (7.30).

Example 7.31 (complete toric lines). Here, we explicitly describe all complete toric orbifolds of dimension 1. Notice that the coarse moduli space of a complete toric line is P1. Denote by x , x the homogeneous coordinates. Let Di be the Cartier divisor O 1 ; xi . Let 1 2 a1, a2 in N>0. Denote by d (resp. m) the greatest common divisor (resp. the lowest common a ; a 1 2 1 multiple) of a1, a2. The Picard group of the root stack D1; D2 =P is isomorphic to Z Z=dZ . Notice that it is not a weighted projective spaceqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in general. As a global a ; a 1 2 1 2 quotient, the stack D1; D2 =P is C 0 = C md where the action is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nf g 2 2 C m C 0 C 0 ; d nf g ! nf g

m=a1 k m=a2 k l; t ; x ; x l t 2 x ; l t 1 x 1 2 7! 1 2 k k 1 where k , k are integers such that 1 2 . 1 2 a b m

Appendix A. Uniqueness of morphisms to separated stacks

We prove Proposition 1.2.

Proposition A.1. Let X and Y be two Deligne-Mumford stacks. Assume that X is normal and Y is separated. Let i : U , X be a dominant open immersion. If F1; F2 : X Y are two morphisms of stacks such that! there exits a 2-arrow b : F i F i then there! exists a 1 ) 2 unique 2-arrow a : F F such that a idi b. 1 ) 2

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238 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Proof. Uniqueness: We ﬁrst assume that X is a scheme, denoted by X, and Y is a global quotient V=G where G is a separated group scheme. Denote by U the scheme U, open dense in X. For i in 1; 2 , the morphism Fi is given by an object xi which is a f g G-torsor pi : Pi X and a G-equivariant morphism Pi V. Let a; a0 : P P be mor- ! ! 1 ! 2 phisms between the objects x1 and x2 such that a p 1 U a0 p 1 U .AsG is separated, we j 1 j 2 have that pi is separated. We deduce that a a . 0 Now we prove the uniqueness of the proposition in the case where Y V=G . Let X X be an e´tale atlas of . By the previous point, we deduce that a X a0 X .AsMorF1; F2 is a sheaf on X, we conclude that a a . j j 0 For the general case, we reduce to the previous by covering Y by global quotients and then we use that Mor F ; F is a sheaf on X. 1 2 Existence: It is enough to do it for an e´tale a‰ne chart of X. By hypothesis, this chart is a disjoint union of a‰ne irreducible normal varieties. Hence, we can assume that X is an a‰ne irreducible normal variety, denoted by X. Denote by U the scheme U open dense in X. The morphism F1 i : U Y, the 2-arrow b and the universal property of the strict ﬁber product give a morphism !f : U U . The existence of a is equivalent ! 0 to the existence of a morphism h : X a X 0 such that p1 h id and h i g f . Denote by D : Y Y Y the diagonal. We can sum up the informations in the following diagram: !

U X

f bh

g p2 U 0 X 0 Y id ! ! rrp D ? ? 1 ? ? i ? F1 F2 ? U? X? Y ? Y: y ! y ! y By definition of the separatedness of Y, we have the D is proper. By [26], Lemma 4.2, we have that D is finite and X 0 is a scheme. We deduce that p1 : X 0 X is finite. The morphism g f : U X is a section of p . By Lemma A.2, we deduce! a morphism ! 0 1 h : X X 0 such that h i g f . This completes the proof. r !

Lemma A.2. Let X 0 be a scheme and X be an irreducible normal variety. Let p : X 0 X be a ﬁnite morphism. Let U , X be an open dense immersion. Let s : U X 0 be a section! of p. Then the section s extends! to a section ss~ : X X . ! ! 0

Proof. Denote by U the closure of the s U in the fiber product U : U X X . 0 0 0 Denote by p : U 0 U and q : U 0 X 0 the morphisms induced by the fiber product ! ! U 0. Looking at the fractional fields, we deduce that the morphisms s : U U0 and p : U U are birational morphisms. Denote by X the closure of U in X!. As the U0 0 0 0 0 morphismj !q is an open embedding, we have that q is dominant. We deduce that U0 p : X X is birational and quasi-finite. As X wouldj be an irreducible normal variety, X0 0 thej Zariski! main theorem implies that p is an isomorphism. Its inverse is the wanted X0 section of p. r j

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 239

Appendix B. Action of a Picard stack

In this appendix, we recall the definition of a Picard stack. Then we define the action of a Picard stack on a stack which extends the definition of Romagny in [32]. In [11], De- finition 6.1, Breen defines the notion of a G-torsor over a stack where G is a Picard stack. Our definition of the action is actually already included in that definition.

To deﬁne the notion of Picard stacks, we do not need the stacks to be algebraic.

Deﬁnition B.1 (Picard stacks [7], Exp. XVIII). Let S be a base scheme. A Picard S-stack G is an S-stack with the following data:

(multiplication) a morphism of S-stacks: m G S G G; ! g ; g g g ; 1 2 7! 1 2 (2-associativity) a 2-arrow y implementing the associativity law:

B:2 yg g g : g g g g g g ; 1; 2; 3 1 2 3 ) 1 2 3 (2-commutativity) a 2-arrow t implementing commutativity:

B:3 tg g : g g g g : 1; 2 1 2 ) 2 1 These data must satisfy the following conditions:

(1) For every chart U and every object g A G U the map mg : G G which multi- ! plies every object by g and every arrow by idg is an isomorphism of stacks.

(2) (Pentagon relation) For every chart U and 4-tuples of objects gi A G U ,we have

B:4 idg yg ; g ; g yg ; g g ; g yg ; g ; g idg yg ; g ; g g yg g ; g ; g : 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

(3) For every chart U and every object g A G U , we have tg; g idg g.

(4) For every chart U and every objects g1; g2 A G U , we have tg ; g tg ; g idg g . 1 2 2 1 2 1

(5) (Hexagon relation) For every chart U and every triple of objects g1, g2, g3 in G U , we have

B:5 yg ; g ; g tg ; g g yg ; g ; g idg tg ; g yg ; g ; g tg ; g idg : 1 2 3 3 1 2 3 1 2 1 3 2 1 3 2 3 1 2 Remark B.6. The pentagon relation establishes the compatibility law between 2- arrows y when expressing the associativity with 4 objects.

The third condition means that every object strictly commutes with itself.

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240 Fantechi, Mann and Nironi, Deligne-Mumford stacks

The last condition states compatibility between the 2-arrow of associativity and the 2-arrow of commutativity.

Remark B.7. It can be proved, see [7], Exp. XVIII, 1.4.4, that the previous deﬁnition is enough to guarantee the existence of a neutral element in the group stack. More precisely it is a couple e; where e : S G is a section and : e e e. A neutral element is unique up to a unique isomorphism.! )

Deﬁnition B.8 (morphisms of Picard stacks [7], Exp. XVIII). Let G; y; t and H; c; r be two Picard S-stacks. A morphism of Picard S-stacks is a morphism of S-stacks F : G H f : F g g F g F g g g G with a 2-arrow g1; g2 1 2 1 2 for any 1, 2 objects of satisfying! the following compatibility conditions: )

For every chart U and every couple of objects g ; g A G U we have 1 2

B:9 rF g ; F g fg ; g fg ; g F tg1; g2 : 1 2 1 2 2 1 For every chart U and every triple of objects g ; g ; g A G U we have 1 2 3

B:10 fg ; g g idF g1 fg ; g F yg1; g2; g3 1 2 3 2 3

cF g ; F g ; F g fg ; g idF g3 fg g ; g : 1 2 3 1 2 1 2 3 Remark B.11. (1) It should be observed that the morphism F maps the pentagon relation (resp. the hexagon relation) for the Picard stack G to the pentagon relation (resp. the hexagon relation) for H.

(2) Denote by eG;G a neutral element of G and eH;H a neutral element of H. 1 G G H The couple F e ; F feG; eG is a neutral element of . By Remark B.7 there exists a 1 2 unique 2-arrow l : F eG eH such that l F G f H l . ) eG; eG

(3) It can be useful to notice that given a : g1 g2 and b : g3 g4 morphisms in G U the following identities involving morphisms holds:) ) 1 F a b f F a F b f : g2; g4 g1; g3 Deﬁnition B.12 (action of a Picard stack). Let G; t; y be a Picard S-stack. Denote by e the neutral section and by the corresponding 2-arrow. Let X be an S-stack. An action of G on X is the following data:

a morphism of S-stack: a G S X X; ! g; x g x; 7! a 2-arrow h: h : e x x; x )

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 241

a 2-arrow s:

sg g x : g g x g g x : 1; 2; 1 2 ) 1 2 These data must satisfy the following conditions:

(1) (Pentagon) For every chart U, every objects g1; g2; g3 A G U and every object x A X U , we have

idg sg ; g ; x sg ; g g ; x yg ; g ; g idx sg ; g ; g x sg g ; g ; x: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 (2) For any chart U and any object x A X U , we have

ide h se e x idx : x ; ; Remark B.13. (1) If the Picard stack is a group-scheme then our deﬁnition of the action is compatible with the one given by Romagny in [32].

(2) Let G; m; y; t be a Picard S-stack. The multiplication m deﬁnes an action of G on itself.

Proposition B.14. Let G1 and G2 be two Picard S-stacks. Let F : G1 G2 be a morphism of Picard stacks with the -arrow f : F g g F g F g . Let! X be an 2 g1; g2 1 2 1 2 S-stack with an action of G given by a; h; s . Then the morphism ) F induces a natural action 2 of G1 on X.

Proof. The natural action is given by aa~; hh~; ss~ where we put: aa~ : a F.

For every object x in X, hh~ : h l idx where l is the 2-arrow deﬁned in x x Remark B.11.

For every couple g ; g of objects of G and every x object of X, 1 2 1 ss~g1; g2; x : sF g1 ; F g2 ; x fg ; g idx . 1 2 It is straightforward but tedious to check that the triple so defined satisfies all the properties in Definition B.12. r

We ﬁnish this section with a proposition about actions on algebraic stacks. We refer to [26], Deﬁnition 12.1, for the notion of e´tale site of a Deligne-Mumford stack.

Proposition B.15. Let X be a smooth Deligne-Mumford stack and G a ﬁnite abelian group. An action of BGonX induces a morphism of sheaves of groups j : G X I gen X on the e´tale site of X. Moreover, as morphism of stacks, jise´tale. !

Proof. We may assume X to be irreducible and d-dimensional. First we produce a stack morphism j : X G I gen X and we prove that j is e´tale. Denote by e : Spec C BG the neutral section.! Denote by D : X X X the diagonal morphism. ! !

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242 Fantechi, Mann and Nironi, Deligne-Mumford stacks

Denote by a : BG X X the action. Using the universal property of the ﬁbered prod-

uct, we have the following ! 2-commutative diagram:

X G

p1 p I X X ! r D B:16 p2

? ?

? D ? id e

X? X ? X ! y ! y

id a id e X BG where p : X G X is the projection. The stack morphism j must be unramified since it is a factor of the! e´tale morphism p : X G X. Since every component of I X has dimension at most d, the stack morphism j is actually! e´tale and its image is contained in I gen X . Now, it remains to prove that j : X I gen X is a morphism of sheaves of groups on the e´tale site of X. The two upper triangles! of diagram (B.16) are strictly commutative since I X is the strict fibered product. This implies that j is a morphism of sheaves of sets over X. Notice that on the e´tale site, the sheaf I X is I gen X . To finish the proof, we need to show that j is a morphism of sheaves of groups. Let us check the compatibility between the composition law in I X and the multiplication of G. This compatibility follows from the existence of a dashed arrow such that the upper square

in the following diagram is strictly commutative:

id m

X G G X G !

j c I X X I X I X ! p2 r p2 ? ? ? p ? I ?X 2 X? y ! y

! j X G p2 where the stack morphism c is the composition law of the inertia stack. The external square of the diagram above is 2-cartesian and the stack morphism id m : X G G X G is the identity on X and the multiplication in G. By the universal property of the strict! ﬁber product, we deduce the dashed arrow such that the upper square is strictly commutative. This ends the proof. r

Appendix C. Stacky version of Zariskis Main Theorem

Here, we prove a stacky version of Zariskis Main Theorem. We did not ﬁnd any reference in the literature for this version.

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Fantechi, Mann and Nironi, Deligne-Mumford stacks 243

Theorem C.1 (Zariskis Main Theorem for stacks). Let X, Y be smooth Deligne- Mumford stacks. Let f : X Y be a representable, birational, quasi-ﬁnite and surjective morphism. Then f is an isomorphism! .

Proof. Let Y Y be an e´tale atlas. Consider the following ﬁber product: !

f X Y ! r ? ? ? f ? X? Y?: y ! y The morphism f : X Y is proper, birational, surjective and quasi-ﬁnite between smooth varieties. Hence, the Zariski! Main Theorem (see for example [29], p. 209) implies that f is an isomorphism. This implies that f : X Y is an isomorphism. r !

References

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244 Fantechi, Mann and Nironi, Deligne-Mumford stacks

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Eingegangen 21. Januar 2009, in revidierter Fassung 19. September 2009

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